Matric Exam Geometry Problem - 1949: Maximizing ∠BEC

The dynamic figure below will help you find how to determine the maximum of ∠BEC. It shows a circle constructed with its centre I on the perpendicular bisector of BC, and tangent to the line segment AD at G. Drag I till the circle passes through B and C.

Now drag E to coincide with G while carefully watching what happens to ∠BEC. What do you notice? Change the shape of the trapezium ABCD, repeat the preceding process, and check your observation again.

1) Can you explain (prove) why the maximum of the angle has to be located at the position of E you identified?

2) Can you figure out a way via construction, not using experimentation and dragging as with this sketch, to precisely locate where the centre I of the circle should be located to determine a maximum for ∠BEC?

 

Maximizing the Angle

Hints:
1) Try using the exterior angle theorem to prove why the maximum of angle BEC is located where it is.
2) Consider the locus of points equidistant from vertex B and line AD and/or the locus of points equidistant from vertex C and line AD.

Check your explorations above, by reading my 2015 paper in At Right Angles and Learning and Teaching Mathematics at "Flashback to the Past: a 1949 Matric Geometry Question".


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Created by Michael de Villiers, 14 August 2015.